# Properties

 Label 2001.2.a.j Level $2001$ Weight $2$ Character orbit 2001.a Self dual yes Analytic conductor $15.978$ Analytic rank $1$ Dimension $7$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2001 = 3 \cdot 23 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2001.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$15.9780654445$$ Analytic rank: $$1$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ Defining polynomial: $$x^{7} - x^{6} - 9 x^{5} + 10 x^{4} + 19 x^{3} - 20 x^{2} - 5 x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{6}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{2} + q^{3} + ( 1 + \beta_{4} + \beta_{5} ) q^{4} + ( -1 + \beta_{3} - \beta_{6} ) q^{5} -\beta_{3} q^{6} + ( -1 - \beta_{2} - \beta_{4} - \beta_{5} ) q^{7} + ( 1 - \beta_{1} + \beta_{2} + \beta_{6} ) q^{8} + q^{9} +O(q^{10})$$ $$q -\beta_{3} q^{2} + q^{3} + ( 1 + \beta_{4} + \beta_{5} ) q^{4} + ( -1 + \beta_{3} - \beta_{6} ) q^{5} -\beta_{3} q^{6} + ( -1 - \beta_{2} - \beta_{4} - \beta_{5} ) q^{7} + ( 1 - \beta_{1} + \beta_{2} + \beta_{6} ) q^{8} + q^{9} + ( -1 - 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{10} + ( -2 - \beta_{5} + \beta_{6} ) q^{11} + ( 1 + \beta_{4} + \beta_{5} ) q^{12} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{5} ) q^{13} + ( -1 + 2 \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{14} + ( -1 + \beta_{3} - \beta_{6} ) q^{15} + ( -2 - \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{16} + ( -2 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{17} -\beta_{3} q^{18} + ( -1 - 3 \beta_{1} + 2 \beta_{3} + \beta_{4} ) q^{19} + ( -2 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{6} ) q^{20} + ( -1 - \beta_{2} - \beta_{4} - \beta_{5} ) q^{21} + ( -2 + \beta_{1} - \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - \beta_{6} ) q^{22} - q^{23} + ( 1 - \beta_{1} + \beta_{2} + \beta_{6} ) q^{24} + ( -1 - \beta_{1} - \beta_{2} + \beta_{4} + \beta_{6} ) q^{25} + ( -1 - 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - \beta_{5} ) q^{26} + q^{27} + ( -4 + \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - 3 \beta_{5} ) q^{28} + q^{29} + ( -1 - 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{30} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} ) q^{31} + ( \beta_{1} - \beta_{4} + \beta_{5} ) q^{32} + ( -2 - \beta_{5} + \beta_{6} ) q^{33} + ( -4 + 2 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{34} + ( 2 + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{35} + ( 1 + \beta_{4} + \beta_{5} ) q^{36} + ( -3 - \beta_{1} - 2 \beta_{2} - 2 \beta_{4} ) q^{37} + ( 1 + 3 \beta_{2} + 2 \beta_{4} + 4 \beta_{5} + \beta_{6} ) q^{38} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{5} ) q^{39} + ( -4 + \beta_{1} - \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{40} + ( 4 + \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + 4 \beta_{5} + \beta_{6} ) q^{41} + ( -1 + 2 \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{42} + ( 2 + 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{43} + ( -3 + \beta_{1} - 2 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} + \beta_{6} ) q^{44} + ( -1 + \beta_{3} - \beta_{6} ) q^{45} + \beta_{3} q^{46} + ( 5 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{47} + ( -2 - \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{48} + ( -\beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{49} + ( 1 + \beta_{1} + 2 \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{50} + ( -2 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{51} + ( -1 + \beta_{2} - 2 \beta_{4} - 2 \beta_{5} ) q^{52} + ( 1 + \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{53} -\beta_{3} q^{54} + ( -\beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{6} ) q^{55} + ( -5 - 3 \beta_{2} + 3 \beta_{3} - 2 \beta_{5} ) q^{56} + ( -1 - 3 \beta_{1} + 2 \beta_{3} + \beta_{4} ) q^{57} -\beta_{3} q^{58} + ( 2 + 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} ) q^{59} + ( -2 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{6} ) q^{60} + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - 3 \beta_{5} + 2 \beta_{6} ) q^{61} + ( -3 - 4 \beta_{2} + 2 \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{6} ) q^{62} + ( -1 - \beta_{2} - \beta_{4} - \beta_{5} ) q^{63} + ( 1 + \beta_{1} - \beta_{3} - 5 \beta_{4} - 4 \beta_{5} + \beta_{6} ) q^{64} + ( 2 - 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + 4 \beta_{6} ) q^{65} + ( -2 + \beta_{1} - \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - \beta_{6} ) q^{66} + ( -4 - \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} + \beta_{6} ) q^{67} + ( -2 \beta_{2} + 5 \beta_{3} - \beta_{5} - 3 \beta_{6} ) q^{68} - q^{69} + ( 7 - 4 \beta_{1} - \beta_{3} + 3 \beta_{4} + 4 \beta_{5} + \beta_{6} ) q^{70} + ( -3 + \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{71} + ( 1 - \beta_{1} + \beta_{2} + \beta_{6} ) q^{72} + ( -6 - 2 \beta_{1} - \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} ) q^{73} + ( 2 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 2 \beta_{6} ) q^{74} + ( -1 - \beta_{1} - \beta_{2} + \beta_{4} + \beta_{6} ) q^{75} + ( 2 - \beta_{1} + \beta_{2} - 3 \beta_{3} + 3 \beta_{5} + \beta_{6} ) q^{76} + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{3} + 4 \beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{77} + ( -1 - 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - \beta_{5} ) q^{78} + ( -6 - 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{79} + ( 3 + \beta_{1} - 2 \beta_{2} - 2 \beta_{5} + 2 \beta_{6} ) q^{80} + q^{81} + ( 1 - 5 \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{82} + ( 1 - 2 \beta_{1} - 5 \beta_{2} + \beta_{3} - \beta_{4} - 3 \beta_{5} + \beta_{6} ) q^{83} + ( -4 + \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - 3 \beta_{5} ) q^{84} + ( 3 \beta_{1} + \beta_{2} - 2 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} - \beta_{6} ) q^{85} + ( -4 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} + \beta_{5} - 2 \beta_{6} ) q^{86} + q^{87} + ( 2 + 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{6} ) q^{88} + ( -1 + 3 \beta_{1} - 2 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - \beta_{6} ) q^{89} + ( -1 - 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{90} + ( -4 - 2 \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} - 3 \beta_{6} ) q^{91} + ( -1 - \beta_{4} - \beta_{5} ) q^{92} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} ) q^{93} + ( 6 - \beta_{1} + \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 6 \beta_{5} - \beta_{6} ) q^{94} + ( -1 + 2 \beta_{1} + \beta_{3} - 4 \beta_{5} - \beta_{6} ) q^{95} + ( \beta_{1} - \beta_{4} + \beta_{5} ) q^{96} + ( -7 + 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{6} ) q^{97} + ( 7 - 5 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + 6 \beta_{5} ) q^{98} + ( -2 - \beta_{5} + \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7q - q^{2} + 7q^{3} + 7q^{4} - 5q^{5} - q^{6} - 5q^{7} + 3q^{8} + 7q^{9} + O(q^{10})$$ $$7q - q^{2} + 7q^{3} + 7q^{4} - 5q^{5} - q^{6} - 5q^{7} + 3q^{8} + 7q^{9} - 11q^{10} - 12q^{11} + 7q^{12} - 13q^{13} - 3q^{14} - 5q^{15} - 13q^{16} - 12q^{17} - q^{18} - 5q^{19} - 8q^{20} - 5q^{21} - q^{22} - 7q^{23} + 3q^{24} - 4q^{25} + 2q^{26} + 7q^{27} - 21q^{28} + 7q^{29} - 11q^{30} - 8q^{31} - 5q^{32} - 12q^{33} - 28q^{34} + 5q^{35} + 7q^{36} - 24q^{37} - 6q^{38} - 13q^{39} - 20q^{40} + 9q^{41} - 3q^{42} - q^{43} - 23q^{44} - 5q^{45} + q^{46} + 27q^{47} - 13q^{48} - 14q^{49} + 7q^{50} - 12q^{51} - 9q^{52} - q^{53} - q^{54} - 11q^{55} - 20q^{56} - 5q^{57} - q^{58} + 8q^{59} - 8q^{60} + q^{61} - 5q^{63} + 3q^{64} + 12q^{65} - q^{66} - 16q^{67} + 15q^{68} - 7q^{69} + 40q^{70} - 13q^{71} + 3q^{72} - 23q^{73} - 8q^{74} - 4q^{75} - 2q^{76} + 13q^{77} + 2q^{78} - 44q^{79} + 30q^{80} + 7q^{81} - 10q^{82} + 21q^{83} - 21q^{84} + 6q^{86} + 7q^{87} + 21q^{88} - 5q^{89} - 11q^{90} - 18q^{91} - 7q^{92} - 8q^{93} + 28q^{94} + 9q^{95} - 5q^{96} - 55q^{97} + 36q^{98} - 12q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - x^{6} - 9 x^{5} + 10 x^{4} + 19 x^{3} - 20 x^{2} - 5 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + 2 \nu^{5} - 7 \nu^{4} - 11 \nu^{3} + 14 \nu^{2} + 10 \nu - 7$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{6} - 2 \nu^{5} + 7 \nu^{4} + 15 \nu^{3} - 10 \nu^{2} - 26 \nu - 1$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{6} - 9 \nu^{4} + \nu^{3} + 20 \nu^{2} - 2 \nu - 5$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{6} - 2 \nu^{5} + 25 \nu^{4} + 9 \nu^{3} - 50 \nu^{2} - 6 \nu + 5$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$($$$$3 \nu^{6} - 2 \nu^{5} - 25 \nu^{4} + 23 \nu^{3} + 46 \nu^{2} - 46 \nu - 5$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} + \beta_{4} + \beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$-\beta_{5} - \beta_{4} + \beta_{3} + 4 \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{6} + 7 \beta_{5} + 5 \beta_{4} - 2 \beta_{3} + 6 \beta_{2} - \beta_{1} + 15$$ $$\nu^{5}$$ $$=$$ $$-\beta_{6} - 10 \beta_{5} - 9 \beta_{4} + 8 \beta_{3} - \beta_{2} + 19 \beta_{1} - 11$$ $$\nu^{6}$$ $$=$$ $$9 \beta_{6} + 44 \beta_{5} + 28 \beta_{4} - 19 \beta_{3} + 34 \beta_{2} - 11 \beta_{1} + 81$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 1.81932 2.13025 −0.325238 −1.64802 0.136094 1.40506 −2.51747
−2.37229 1.00000 3.62775 2.09663 −2.37229 −1.30993 −3.86149 1.00000 −4.97382
1.2 −1.55072 1.00000 0.404722 −0.920116 −1.55072 −2.53796 2.47382 1.00000 1.42684
1.3 −1.49169 1.00000 0.225141 −2.94997 −1.49169 1.89422 2.64754 1.00000 4.40045
1.4 −0.865893 1.00000 −1.25023 1.66575 −0.865893 −0.715958 2.81435 1.00000 −1.44236
1.5 1.17088 1.00000 −0.629030 0.418864 1.17088 1.98148 −3.07829 1.00000 0.490441
1.6 1.75752 1.00000 1.08886 −2.67591 1.75752 0.0258051 −1.60134 1.00000 −4.70295
1.7 2.35219 1.00000 3.53279 −2.63525 2.35219 −4.33765 3.60540 1.00000 −6.19860
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$23$$ $$1$$
$$29$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2001.2.a.j 7
3.b odd 2 1 6003.2.a.i 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2001.2.a.j 7 1.a even 1 1 trivial
6003.2.a.i 7 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2001))$$:

 $$T_{2}^{7} + T_{2}^{6} - 10 T_{2}^{5} - 10 T_{2}^{4} + 29 T_{2}^{3} + 29 T_{2}^{2} - 24 T_{2} - 23$$ $$T_{5}^{7} + 5 T_{5}^{6} - 3 T_{5}^{5} - 40 T_{5}^{4} - 15 T_{5}^{3} + 87 T_{5}^{2} + 36 T_{5} - 28$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-23 - 24 T + 29 T^{2} + 29 T^{3} - 10 T^{4} - 10 T^{5} + T^{6} + T^{7}$$
$3$ $$( -1 + T )^{7}$$
$5$ $$-28 + 36 T + 87 T^{2} - 15 T^{3} - 40 T^{4} - 3 T^{5} + 5 T^{6} + T^{7}$$
$7$ $$-1 + 37 T + 68 T^{2} - 3 T^{3} - 38 T^{4} - 5 T^{5} + 5 T^{6} + T^{7}$$
$11$ $$23 + 90 T - 2 T^{2} - 150 T^{3} - 9 T^{4} + 39 T^{5} + 12 T^{6} + T^{7}$$
$13$ $$-1867 - 4597 T - 4314 T^{2} - 1849 T^{3} - 294 T^{4} + 27 T^{5} + 13 T^{6} + T^{7}$$
$17$ $$12751 + 14263 T + 3350 T^{2} - 1068 T^{3} - 443 T^{4} - 2 T^{5} + 12 T^{6} + T^{7}$$
$19$ $$-7244 - 3664 T + 3993 T^{2} + 909 T^{3} - 306 T^{4} - 62 T^{5} + 5 T^{6} + T^{7}$$
$23$ $$( 1 + T )^{7}$$
$29$ $$( -1 + T )^{7}$$
$31$ $$-8156 + 6494 T + 7247 T^{2} + 278 T^{3} - 515 T^{4} - 54 T^{5} + 8 T^{6} + T^{7}$$
$37$ $$85172 + 64270 T - 4649 T^{2} - 6432 T^{3} - 483 T^{4} + 146 T^{5} + 24 T^{6} + T^{7}$$
$41$ $$-1065508 + 685908 T - 136507 T^{2} + 1335 T^{3} + 2446 T^{4} - 187 T^{5} - 9 T^{6} + T^{7}$$
$43$ $$-15268 + 7782 T + 13761 T^{2} + 2793 T^{3} - 380 T^{4} - 115 T^{5} + T^{6} + T^{7}$$
$47$ $$-10031 - 4296 T + 12110 T^{2} - 3545 T^{3} - 326 T^{4} + 226 T^{5} - 27 T^{6} + T^{7}$$
$53$ $$596 + 1262 T - 955 T^{2} - 525 T^{3} + 506 T^{4} - 105 T^{5} + T^{6} + T^{7}$$
$59$ $$128 + 128 T - 640 T^{2} + 192 T^{3} + 192 T^{4} - 36 T^{5} - 8 T^{6} + T^{7}$$
$61$ $$-36740 - 16402 T + 7949 T^{2} + 3219 T^{3} - 263 T^{4} - 134 T^{5} - T^{6} + T^{7}$$
$67$ $$124927 + 60419 T - 16308 T^{2} - 14086 T^{3} - 2557 T^{4} - 80 T^{5} + 16 T^{6} + T^{7}$$
$71$ $$-16 + 4 T + 367 T^{2} - 479 T^{3} - 201 T^{4} + 24 T^{5} + 13 T^{6} + T^{7}$$
$73$ $$4228 + 55498 T - 11447 T^{2} - 20226 T^{3} - 3953 T^{4} - 59 T^{5} + 23 T^{6} + T^{7}$$
$79$ $$127580 - 162018 T - 78309 T^{2} + 772 T^{3} + 4205 T^{4} + 692 T^{5} + 44 T^{6} + T^{7}$$
$83$ $$-212788 + 184452 T - 32991 T^{2} - 7657 T^{3} + 2362 T^{4} - 41 T^{5} - 21 T^{6} + T^{7}$$
$89$ $$-18865 - 48069 T - 18732 T^{2} + 11085 T^{3} - 274 T^{4} - 195 T^{5} + 5 T^{6} + T^{7}$$
$97$ $$-1979204 - 998308 T - 62983 T^{2} + 46413 T^{3} + 11671 T^{4} + 1170 T^{5} + 55 T^{6} + T^{7}$$